Relations between B, H, and M

The magnetization defines the auxiliary magnetic field H as

${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H+M} )}$ (SI units)

${\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {M} }$ (Gaussian units)

which is convenient for various calculations. The vacuum permeability μ₀ is, approximately, 4π×10⁻⁷ V·s/(A·m) (in SI units).

A relation between M and H exists in many materials. In diamagnets and paramagnets, the relation is usually linear:

${\displaystyle \mathbf {M} =\chi \mathbf {H} ,\,\mathbf {B} =\mu \mathbf {H} =\mu _{0}(1+\chi )\mathbf {H} ,}$

where χ is called the volume magnetic susceptibility, and μ is called the magnetic permeability of the material. The magnetic potential energy per unit volume (i.e. magnetic energy density) of the paramagnet (or diamagnet) in the magnetic field is:

${\displaystyle -\mathbf {M} \cdot \mathbf {B} =-\chi \mathbf {H} \cdot \mathbf {B} =-{\frac {\chi }{1+\chi }}{\frac {\mathbf {B} ^{2}}{\mu _{0}}},}$

the negative gradient of which is the magnetic force on the paramagnet (or diamagnet) per unit volume (i.e. force density).

In diamagnets (${\displaystyle \chi <0}$) and paramagnets (${\displaystyle \chi >0}$), usually ${\displaystyle |\chi |\ll 1}$, and therefore ${\displaystyle \mathbf {M} \approx \chi {\frac {\mathbf {B} }{\mu _{0}}}}$.

In ferromagnets there is no one-to-one correspondence between M and H because of magnetic hysteresis.

Magnetic polarization

Alternatively to the magnetization, one can define the magnetic polarization, I (often the symbol J is used, not to be confused with current density).^[3]

${\displaystyle \mathbf {B} =\mu _{0}\mathbf {H} +\mathbf {I} }$ (SI units).

This is by direct analogy to the electric polarization, ${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }$. The magnetic polarization thus differs from the magnetization by a factor of μ₀:

${\displaystyle \mathbf {I} =\mu _{0}\mathbf {M} }$ (SI units).

Whereas magnetization is measured typically in amperes/meter, the magnetic polarization is measured in teslas.

Magnetization current

When the microscopic currents induced by the magnetization (black arrows) do not balance out, bound volume currents (blue arrows) and bound surface currents (red arrows) appear in the medium.

The magnetization M makes a contribution to the current density J, known as the magnetization current.^[4]

${\displaystyle \mathbf {J} _{\mathrm {m} }=\nabla \times \mathbf {M} }$

and for the bound surface current:

${\displaystyle \mathbf {K} _{\mathrm {m} }=\mathbf {M} \times \mathbf {\hat {n}} }$

so that the total current density that enters Maxwell's equations is given by

${\displaystyle \mathbf {J} =\mathbf {J} _{\mathrm {f} }+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}$

where J_f is the electric current density of free charges (also called the free current), the second term is the contribution from the magnetization, and the last term is related to the electric polarization P.

Magnetostatics

In the absence of free electric currents and time-dependent effects, Maxwell's equations describing the magnetic quantities reduce to

${\displaystyle {\begin{aligned}\mathbf {\nabla \times H} &=\mathbf {0} \\\mathbf {\nabla \cdot H} &=-\nabla \cdot \mathbf {M} \end{aligned}}}$

These equations can be solved in analogy with electrostatic problems where

${\displaystyle {\begin{aligned}\mathbf {\nabla \times E} &=\mathbf {0} \\\mathbf {\nabla \cdot E} &={\frac {\rho }{\epsilon _{0}}}\end{aligned}}}$

In this sense −∇⋅M plays the role of a fictitious "magnetic charge density" analogous to the electric charge density ρ; (see also demagnetizing field).